Optimal. Leaf size=40 \[ -\frac {a^3}{4 x^4}+\frac {3}{4} a b^2 x^4+\frac {b^3 x^8}{8}+3 a^2 b \log (x) \]
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Rubi [A]
time = 0.02, antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {272, 45}
\begin {gather*} -\frac {a^3}{4 x^4}+3 a^2 b \log (x)+\frac {3}{4} a b^2 x^4+\frac {b^3 x^8}{8} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 272
Rubi steps
\begin {align*} \int \frac {\left (a+b x^4\right )^3}{x^5} \, dx &=\frac {1}{4} \text {Subst}\left (\int \frac {(a+b x)^3}{x^2} \, dx,x,x^4\right )\\ &=\frac {1}{4} \text {Subst}\left (\int \left (3 a b^2+\frac {a^3}{x^2}+\frac {3 a^2 b}{x}+b^3 x\right ) \, dx,x,x^4\right )\\ &=-\frac {a^3}{4 x^4}+\frac {3}{4} a b^2 x^4+\frac {b^3 x^8}{8}+3 a^2 b \log (x)\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 40, normalized size = 1.00 \begin {gather*} -\frac {a^3}{4 x^4}+\frac {3}{4} a b^2 x^4+\frac {b^3 x^8}{8}+3 a^2 b \log (x) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.16, size = 35, normalized size = 0.88
method | result | size |
default | \(-\frac {a^{3}}{4 x^{4}}+\frac {3 a \,b^{2} x^{4}}{4}+\frac {b^{3} x^{8}}{8}+3 a^{2} b \ln \left (x \right )\) | \(35\) |
norman | \(\frac {-\frac {1}{4} a^{3}+\frac {1}{8} b^{3} x^{12}+\frac {3}{4} a \,b^{2} x^{8}}{x^{4}}+3 a^{2} b \ln \left (x \right )\) | \(37\) |
risch | \(\frac {b^{3} x^{8}}{8}+\frac {3 a \,b^{2} x^{4}}{4}+\frac {9 a^{2} b}{8}-\frac {a^{3}}{4 x^{4}}+3 a^{2} b \ln \left (x \right )\) | \(41\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.31, size = 36, normalized size = 0.90 \begin {gather*} \frac {1}{8} \, b^{3} x^{8} + \frac {3}{4} \, a b^{2} x^{4} + \frac {3}{4} \, a^{2} b \log \left (x^{4}\right ) - \frac {a^{3}}{4 \, x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 38, normalized size = 0.95 \begin {gather*} \frac {b^{3} x^{12} + 6 \, a b^{2} x^{8} + 24 \, a^{2} b x^{4} \log \left (x\right ) - 2 \, a^{3}}{8 \, x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.05, size = 37, normalized size = 0.92 \begin {gather*} - \frac {a^{3}}{4 x^{4}} + 3 a^{2} b \log {\left (x \right )} + \frac {3 a b^{2} x^{4}}{4} + \frac {b^{3} x^{8}}{8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.56, size = 46, normalized size = 1.15 \begin {gather*} \frac {1}{8} \, b^{3} x^{8} + \frac {3}{4} \, a b^{2} x^{4} + \frac {3}{4} \, a^{2} b \log \left (x^{4}\right ) - \frac {3 \, a^{2} b x^{4} + a^{3}}{4 \, x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.03, size = 34, normalized size = 0.85 \begin {gather*} \frac {b^3\,x^8}{8}-\frac {a^3}{4\,x^4}+\frac {3\,a\,b^2\,x^4}{4}+3\,a^2\,b\,\ln \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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